3. In an arbitrary RC circuit with constant emf E, use the method of sepa- ration of variables to derive the formula Q(t) = Ae-t/RC + EC for the charge on the capacitor, where A is an arbitrary constant. If Q(0) = Qo, what is A?
3. In an arbitrary RC circuit with constant emf E, use the method of sepa- ration of variables to derive the formula Q(t) = Ae-t/RC + EC for the charge on the capacitor, where A is an arbitrary constant. If Q(0) = Qo, what is A?
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Please do #3
Transcribed Image Text:the solution is continuous, we impose Q1(2)
Q1 equation; A
resp
is an arbitrary
22(2), which gives A = (1/2) (e* - 1). So
E(t) = {(¹-1) e-21, 0≤t≤2,
t> 2.
EXERCISES
1. Write the equation that governs an RC circuit with a 12-volt battery, taking
R = 1 and C= . Determine the equilibrium solution and its stability. If
Q(0) = 5, find a formula for Q(t). Find the current I(t). Plot the charge
and the current on the same set of axes.
2. An aging battery generating 200e-5 volts is connected in series with a
20 ohm resistor, and a 0.01 farad capacitor. Assuming q = 0 at t = 0,
find the charge and current for all t > 0. Show that the charge reaches a
maximum and find the time it is reached.
3. In an arbitrary RC circuit with constant emf E, use the method of sepa-
ration of variables to derive the formula
Q(t) = Ae-t/RC
+ EC
for the charge on the capacitor, where A is an arbitrary constant. If Q(0) =
Qo, what is A?
4. An RCL circuit with an applied emf given by E(t) has initial charge Q(0) =
2o and initial current I(0) = Io. What is I'(0)? Write the circuit equation
and the initial conditions in terms of current I(t). Hint: Use Kirchhoff's
law in the form LI' + RI+Q/C = E(t).
5. Write the RCL circuit equation with the voltage V.(t) on the capacitor as
the unknown state function.
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